Dynamic compression and weak shock formation in an inert gas due to fast piston acceleration

Abstract

Unsteady gasdynamic concepts are used to model the piston-driven compression of a confined gas. Perturbation methods, based on the limit of small piston Mach number, are used to construct solutions. The piston Mach number increases smoothly from zero to a maximum value, M_p = O(10⁻²) during an acoustic time period t_a* = O(10⁻⁴ s). A linear a coustic field is generated and is represented in terms of an infinite series of Fourier spatial modes. During the longer piston time period t_p* = O(10⁻² s) the piston moves at constant speed. A multiple-timescale formulation is used to separate the instantaneous acoustic field from the accumulated bulk response of the gas to piston compression. The latter is found to be identical to the classical quasi-static results from equilibrium thermodynamic calculations. Nonlinear effects become important on the piston timescale. Modal interactions are represented by a system of coupled, nonlinear ordinary differential equations for the time-dependent Fourier coefficients. A numerical solution for this system describes the wavefront steepening to form a weak shock and its propagation back and forth repeatedly inside the cylinder.